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Remote Sensing LaboratoryDept. of Information Engineering and Computer Science

University of TrentoVia Sommarive, 14, I-38123 Povo, Trento, Italy

Remote Sensing LaboratoryDept. of Information Engineering and Computer Science

University of TrentoVia Sommarive, 14, I-38123 Povo, Trento, Italy

Begüm Demir

Digital Signal ProcessingLecture 4

Digital Signal ProcessingLecture 4

E-mail: [email protected] page: http://rslab.disi.unitn.it

University of Trento, Italy 2B. Demir

Discrete TimeSystem

[ ] [ ]

1 0[ ]

0 0[ ] [ ]

n

l

y n x l

nh n

nh n u n

0

0

[ ] [ ]

[ ] [ ]

[ ] [ ]

k

k

y n x n k

h n n k

h n u n

1 2 3

1 2 3

[ ] [ ] [ 1] [ 2][ ] [ ] [ 1] [ 2]

y n a x n a x n a x nh n a n a n a n

[ ]h n[ ]n

Linear Time-Invariant (LTI) Systems

Impulse Response


Linear Time-Invariant (LTI) Systems

[ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ]k k

x n y n

n h nn k h n k

x k n k x k h n k

[ ] [ ] [ ]k

y n x k h n k

Convolution sum:

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Linear Time-Invariant Systems

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LTI Systems: Convolution


Example

1 1

[ ] (0.2) [ ][ ] (0.6) [ ]

[ ] [ ] [ ]?

[ ] 2.5(0.6 0.2 ) [ ]

n

n

n n

x n u nh n u n

y n x n h n

y n u n


Convolution Sum-Example

y[n]?

The output of an LTI system can be obtained as the superposition of responsesto individual samples of the input. This approach is shown to estimate y[n] inthe case of x[n] and h[n] given in the following:

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Convolution Sum-Example-cont

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Example

y[n]?


Example-cont


The output sequence y[n] can be also obtained by multiplying the inputsequence (expressed as a function of k) by the sequence whose valuesare h[n − k], −∞ < k < ∞ for any fixed value of n, and then summing all thevalues of the products x[k]h[n−k], with k a counting index in thesummation process.

Therefore, the operation of convolving two sequences involves doing thecomputation specified by

for each value of n, thus generating the complete output sequence y[n], −∞ <n < ∞.

k

knhkxny ][][][

LTI Systems-Convolution

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The process of computing the convolution between x[n] and h[n] involvesthe following steps:

1. Express x[n] and h[n] as a function of k and obtain x[k] and h[k]

2. Obtain h[-k] from h[k].

3. Shift h[-k] by n0 to the right if n is positive to obtain h[n0 – k] (shifth[- k] by n0 to the left if n is negative).

4. Multiply x [k] by h[n0 -k] to obtain the product sequence.

5. Sum all the values of the product sequence to obtain the value of theoutput at time n= n0.

The steps from 3 to 5 should be repeated for each values of n0 (i.e., forany fixed value of n).

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k

knhkxny ][][][

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For each n value, estimate h[n-k], and then multiply with x[k] to estimate y[n].



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[ ] [ ]* [ ] [ ] [ ]k

y n x n h n x k h n k

0 1 2 3 4 5 6k

x[k]

0 1 2 3 4 5 6kh[k]

0 1 2 3 4 5 6kh[0k]



0 1 2 3 4 5 6k

x[k]

0 1 2 3 4 5 6kh[0k]

0 1 2 3 4 5 6kh[1k]

compute y[0]

compute y[1]

How to compute y[n]?


For each n value, estimate h[n-k], and then multiply with x[k] to estimate y[n].

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Example

[ ] 1 2 3 4 5

[ ] 1 2 1 [ ] ?

x n

h n y n


[ ] 1 1 1 1 1 1

[ ] 1 0.5 0.25 0.125

[ ]?

h n

x n

y n

Example


Convolution Features

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1 2 3 1 2 3associative: [ ] [ ] [ ] [ ] [ ] [ ]x n x n x n x n x n x n

1 2 3 1 2 1 3distributive: [ ] [ ] [ ] [ ] [ ] [ ] [ ]x n x n x n x n x n x n x n

1 2 2 1commutative: [ ] [ ] [ ] [ ]x n x n x n x n

1 1[ ] [ ] [ ]x n n x n 1 2

1 2

if [ ] has samples and [ ] samples, [ ] [ ] [ ] will have -1 samples. x n m x n k

c n x n x n m k

1 2 2

1 2 2

[ ] [ ] [ ][ ] [ ] [ ]

x n x n c nx n m x n k c n m k

1 1[ ] [ ] [ ]x n n x n


LTI Systems-Parallel Connection of Systems

x[n] x[n]y[n] y[n]

h2[n]

h1[n]

h1[n]+h2[n]

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LTI Systems-Cascade Connection of Systems

x[n] x[n]y[n] y[n]h2[n]h1[n] h1[n]*h2[n]

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LTI Systems-Inverse Systems

x[n] x[n]y[n]h'[n]h[n]

[ ] '[ ] [ ]h n h n n

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Example-Inverse Systems

0

0

'[ ] [ ] [ ]

'[ ] [ ]

'[ ] '[ 1] '[ 2] ... [ ]'[0] 1 '[1] '[0] 0 '[1] 1'[2] 0, '[ ] 0, 1'[ ] [ ] [ 1]

k

k

h n n k n

h n k n

h n h n h n nh h h hh h n nh n n n

0

0

[ ] [ ]

[ ] [ ] [ ]

'[ ] ?

k

k

y n x n k

h n n k u n

h n

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Stability of Systems

[ ]k

h k

This is due to the fact that:

[ ] [ ] [ ] ,

[ ] [ ] [ ] [ ] [ ]

if [ ] s bounded ( [ ] ),so that

[ ] [ ] , [ ]

yk

k k

x

xk k

y n h k x n k B

y n h k x n k h k x n k

x n i x n B

y n B h k h k

All LTI systems are BIBO stable if and only if h[n] is absolutely summable:

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Stability of Systems-Example

[ ] [ ] is stable?nh n u n

0

[ ]

if 1 the system is stable

if 1 the system is NOT stable

k

k k

h k a

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[ ] [ ]?

1 n 0=

0 n



2

11 2

1 21 2

1[ ] [ ]?1

1 1=

0 otherwise

M

k M

h n n kM M

M n MM M

The system is stable

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Causality of Systems

[ ] 0 for 0h n n

A LTI system is causal if an only if

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0[ ] [ ]Is [ ] causal?y n x n n

h n

Since h[n] includes only one sample at n=n0 , h[n] is causal.It is stable because it is absolutely summable.

0[ ] [ ]h n n n

[ ] [ 1] [ ]Is [ ] causal?y n x n x n

h n

Since h[n] includes sample at n=-1 , h[n] is not causal, however it is stable because it is absolutely summable.

[ ] [ 1] [ ]h n n n

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1

0

1[ ] [ ]

Is [ ] causal?

N

k

y n x n kN

h n

Since h[n] does not include samples when at n



[ ] [ ]

Is [ ] causal?

n

k

y n x k

h n

Since h[n] does not include samples when at n


FIR and IIR Systems

Finite Impulse Response (FIR) Systems: Systems with only a finite length of nonzero values in h[n] are

called FIR systems.

Examples: Ideal delay, moving average filter…

FIR systems are always STABLE

Infinite Impulse Response (IIR) Systems: Systems with infinite length of nonzero values in h[n] are called

IIR systems.

Examples:Accumulator, filters …

IR systems can be STABLE or UNSTABLE

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FIR Systems: IIR Systems:

0

1

0

[ ] [ ]

[ ] [ 1] [ ]

1[ ] [ ]N

k

h n n n

h n n n

h n n kN

[ ] [ ]

[ ] [ ]n

h n u n

h n a u n

FIR and IIR Systems

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